The variable bound conjecture for minimally unsatisfiable $k$-DNF sets

Let $k$ be a fixed positive integer, and let ${\mathbb{D}}$ be a minimally unsatisfiable set of $k$-DNF formulas. Write $\lvert{\mathbb{D}}\rvert$ for the number of formulas in ${\mathbb{D}}$. The number of variables occurring in ${\mathbb{D}}$ is at most

$$\mathrm{O}\bigl(\lvert{\mathbb{D}}\rvert^k\bigr).$$

Variable bound conjecture. Suppose that ${\mathbb{D}}$ is a minimally unsatisfiable $k$-DNF set for some arbitrary but fixed positive integer $k$. Then the number of variables in ${\mathbb{D}}$ is at most $\mathrm{O}\bigl(\lvert{\mathbb{D}}\rvert^k\bigr)$. This conjecture would close the gap between the known lower bound $\Omega(m^k)$ and upper bound $\mathrm{O}((mk)^{k+1})$ for the number of variables in a minimally unsatisfiable $k$-DNF set with $m$ formulas; the exponent is expected to decrease from $k+1$ to $k$.